On some spaces of functions with bounded derivatives between manifolds (Q1906543)

Let \(M\) be a \(C^{1}\)-compact manifold with or without boundary, and let \(BV(M,S^{1}) = \{ u \in L^{1}(M,S^{1})\), \(\nabla u \in M^{1}(M)\}\), where \(M^{1}(M)\) denotes the space of all bounded measures on \(M\). Here the author shows the weak density result of \(C^{\infty}(M, S^{1})\) in \(BV(M, S^{1})\). Moreover in the case when \(M = B^{2}\), she gives a characterization of \(BV(B^{2}, S^{1})\) as \(\{u \in L^{1}(B^{2}, S^{1}) :{ 1 \over 2}(u_{i,j}+ u_{j,i}) \in M^{1}(B^{2})\), \(i,j \in\{1, 2 \} \}.\)